$\mathbb R$-linear conjugation problem on the unit circle in the parabolic case

A solution to the $\mathbb R$-linear conjugation problem (Markushevich boundary value problem) on the unit circle was proposed. This problem is analogous to the vector-matrix Riemann boundary value problem with the coefficient degenerating in the parabolic case (the coefficient is a triangular matrix function). A complete description of the factorization of the matrix coefficient was provided. Its partial indices were calculated. The method used is based on G.N. Chebotarev’s algorithm and has been developed in a series of author's articles. The resulting factorization confirms the solvability of the $\mathbb R$-linear conjugation problem on the unit circle in the parabolic case.